Question

Degree of degeneracy will be large when:

• A. temperature is high, particle density is large
• B. temperature is high, particle density is small
• C. temperature is low, particle density is small
• D. temperature is low, particle density is large

Hint:

Degeneracy in this context means "quantum-ness." Quantum effects dominate when particles are cold (slow-moving, large de Broglie wavelength) and crowded (small inter-particle spacing).

Answer 1

The Correct Option is D

A quantum gas is said to be "degenerate" when quantum effects (due to the Pauli exclusion principle for fermions or Bose-Einstein condensation for bosons) become important. This happens when the thermal de Broglie wavelength of the particles becomes comparable to or larger than the average inter-particle separation. The thermal de Broglie wavelength is \(\lambda_{th} = \frac{h}{\sqrt{2\pi m k_B T}}\). It is large when the temperature \(T\) is low. The average inter-particle separation is related to the particle density \(n = N/V\). It is small when the particle density \(n\) is large. Therefore, the system becomes degenerate (the degree of degeneracy is large) when the temperature is low and the particle density is high. This forces particles to occupy higher energy states (for fermions) or crowd into the ground state (for bosons), which are distinctly quantum behaviors.